Problem 28 - Precalculus Mathematics for Calculus

Question

Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t .\) $$t=\frac{5 \pi}{3}$$

Step-by-Step Solution

Answer

The terminal point is \( P \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \).

Step 1: Understand the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane. The angle \( t \) in radians corresponds to a point \( P(x, y) \) on the circle.

Step 2: Determine Reference Angle

The given angle is \( t = \frac{5 \pi}{3} \). Since this angle is greater than \( 2\pi \), we find a coterminal angle by subtracting \( 2\pi \) (since it is equivalent to a complete circle). We have:\[ t_{coterminal} = \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3} \].Since we want a positive angle between 0 and \(2\pi\), we add \( 2\pi \):\[ \theta = -\frac{\pi}{3} + 2\pi = \frac{5\pi}{3} \].The reference angle, however, can be found by observing that \( \frac{5\pi}{3} \) itself is already a valid equivalent angle for evaluation.

Step 3: Evaluate Trigonometric Values

The reference angle for \( \frac{5\pi}{3} \) is \( \frac{\pi}{3} \). In the unit circle, the cosine of an angle gives the x-coordinate, and the sine of an angle gives the y-coordinate. Hence:For \( \frac{\pi}{3} \) (60 degrees), \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \) and \( \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \).

Step 4: Determine the Correct Quadrant

\( \frac{5\pi}{3} \) is located in the fourth quadrant. In this quadrant, sine values are negative and cosine values are positive. Therefore:- The x-coordinate (cosine) remains positive: \( x = \frac{1}{2} \).- The y-coordinate (sine) becomes negative: \( y = -\frac{\sqrt{3}}{2} \).

Step 5: Write the Terminal Point

Combining the x and y coordinates, the terminal point \( P(x, y) \) on the unit circle for \( t = \frac{5\pi}{3} \) is:\[ P \left( \frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \].

Key Concepts

unit circle

The unit circle is an essential concept in trigonometry and helps in understanding angles and trigonometric functions. Imagine a circle with a radius of one unit centered at the origin of a coordinate plane, 0,0. Every point on this circle is described by a pair of coordinates (x,y), which are based on the trigonometric functions of a given angle.
The equation of the unit circle is \( x^2 + y^2 = 1 \). Here, '\(x\)' represents the cosine of the angle, and '\(y\)' represents the sine. The circle is a complete 360-degree rotation, marked by radians.

• In trigonometry, using the unit circle, we can not only find sine and cosine values but also tangent and their inverses.
• The unit circle simplifies understanding of periodic trigonometric functions.

From learning the unit circle, you'll be able to solve problems involving angles, find coordinates, and calculate trigonometric functions with ease.

radian measure

Radian measure is a way of expressing angles. It's based on the radius of a circle. Unlike degrees, where a circle is divided into 360 parts, radians use the arc length and radius. A full circle is equal to \(2\pi\) radians. This stems from the fact that the circumference of a circle is \(2\pi r\), where 'r' is the radius.
In simpler terms:

• A full circle is \(2\pi\) radians.
• A half circle, or semicircle, is \(\pi\) radians.

Radians are more prevalent in advanced math because they provide a more natural way of relating angle to space and distance.When using radians, it's crucial to be comfortable converting between radians and degrees:

• Multiply degrees by \(\frac{\pi}{180}\) to obtain radians.

Understanding radians aids in interpreting unit circle problems and evaluating trigonometric functions that deal with periodic cycles.

reference angle

A reference angle is a staple in trigonometry for simplifying calculations on the unit circle. It is the smallest angle that the terminal side of a given angle makes with the x-axis. It is always positive and always less than \(\pi/2\) radians, (90\text{ degrees}).To find a reference angle:

• For angles in the first quadrant, the reference angle is the angle itself.
• In the second quadrant, subtract the angle from \(\pi\) (or 180\text{ degrees}).
• For angles in the third quadrant, subtract \(\pi\) (or 180\text{ degrees}) from the angle.
• Lastly, in the fourth quadrant, subtract the angle from \(2\pi\) (or 360\text{ degrees}).

A reference angle helps by allowing one to use known trigonometric values, particularly basic unit circle angles, to derive those of more complex angles.

quadrants in trigonometry

In trigonometry, the coordinate plane is divided into four quadrants, each representing different sign conventions for trigonometric functions:

• **First Quadrant (I):** Both sine and cosine values are positive here.
• **Second Quadrant (II):** Sine is positive, while cosine is negative.
• **Third Quadrant (III):** Both sine and cosine values are negative.
• **Fourth Quadrant (IV):** Sine is negative while cosine is positive.

These quadrants are determined by the angle's position after being measured from the positive x-axis. Knowing which quadrant an angle is in helps identify the signs of the corresponding (x, y) values on the unit circle.It's vital to quickly identify which quadrant an angle resides in, as this clarifies the behavior of trigonometric functions:
- Angles from 0 to \(\pi/2\) place you in the first quadrant.
- Those from \(\pi/2\) to \(\pi\) are in the second.
- Angles stretching from \(\pi\) to \(3\pi/2\) situate you in the third quadrant.
- Finally, from \(3\pi/2\) to \(2\pi\).This classification is fundamental when solving trigonometry problems as it enables correct evaluation of functions.